Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Upper bounds for ergodic sums of infinite measure preserving transformations


Authors: Jon Aaronson and Manfred Denker
Journal: Trans. Amer. Math. Soc. 319 (1990), 101-138
MSC: Primary 28D05; Secondary 60F15
DOI: https://doi.org/10.1090/S0002-9947-1990-1024766-3
MathSciNet review: 1024766
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For certain conservative, ergodic, infinite measure preserving transformations $ T$ we identify increasing functions $ A$, for which

$\displaystyle \limsup \limits_{n \to \infty } \frac{1} {{A(n)}}\sum\limits_{k = 1}^n {f \circ } {T^k} = \int_X {fd\mu } \quad {\text{a}}{\text{.e}}{\text{.}}$

holds for any nonnegative integrable function $ f$. In particular the results apply to some Markov shifts and number-theoretic transformations, and include the other law of the iterated logarithm.

References [Enhancements On Off] (What's this?)

  • [1] J. Aaronson, Rational ergodicity and a metric invariant for Markov shifts, Israel J. Math. 27 (1977), 93-123. MR 0584018 (58:28424)
  • [2] -, The asymptotic distributional behaviour of transformations preserving infinite measure, J. Analyse Math. 39 (1981), 203-234. MR 632462 (82m:28030)
  • [3] -, An ergodic theorem with large normalizing constants, Israel J. Math. 38 (1981), 182-188. MR 605376 (83f:28014)
  • [4] -, Random $ f$-expansions, Ann. Probab. 14 (1986), 1037-1057. MR 841603 (87k:60057)
  • [5] R. L. Adler, Continued fractions and Bernoulli trials, Ergodic Theory (J. Moser, E. Phillips and S. Varadhan, Eds.), Courant Inst. Math. Sci., New York, 1975. MR 0486431 (58:6177)
  • [6] R. L. Adler and B. Weiss, The ergodic infinite measure preserving transformation of Boole, Israel J. Math. 16 (1973), 263-278. MR 0335751 (49:531)
  • [7] K. L. Chung and P. Erdàs, On the applications of the Borel-Cantelli lemma, Trans. Amer. Math. Soc. 72 (1952), 179-186. MR 0045327 (13:567b)
  • [8] K. L. Chung and G. A. Hunt, On the zeroes of $ \sum\nolimits_1^n { \pm 1} $, Ann. of Math. (2) 50 (1949), 385-400. MR 0029488 (10:613e)
  • [9] D. A. Darling and M. Kac, On occupation times for Markov processes, Trans. Amer. Math. Soc. 84 (1957), 444-458. MR 0084222 (18:832a)
  • [10] M. D. Donsker and S. R. S. Varadhan, On laws of the iterated logarithm for local times, Comm. Pure Appl. Math. 30 (1977), 707-753. MR 0461682 (57:1667)
  • [11] E. Hopf, Ergodentheorie, Chelsea, New York, 1948.
  • [12] I. B. Ibragimov and Y. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff, Groningen, 1971. MR 0322926 (48:1287)
  • [13] N. C. Jain, Some limit theorems for general Markov processes, Z. Wahrsch. Verw. Gebiete 6 (1966), 206-233. MR 0216580 (35:7411)
  • [14] N. C. Jain and W. E. Pruitt, An invariance principle for the local time of a recurrent random walk, Z. Wahrsch. Verw. Gebiete 66 (1984), 141-156. MR 743090 (86i:60090)
  • [15] M. Kac, Of the notion of recurrence in discrete stochastic processes, Bull. Amer. Math. Soc. 53 (1947), 1002-1010. MR 0022323 (9:194a)
  • [16] S. Kakutani, Induced measure preserving transformations, Proc. Imp. Acad. Sci. Tokyo 19 (1943), 635-641. MR 0014222 (7:255f)
  • [17] H. Kesten, An iterated logarithm law for local time, Duke Math. J. 32 (1965), 447-456. MR 0178494 (31:2751)
  • [18] T. Li, F. Schweiger, The generalized Boole transformation is ergodic, Manuscripta Math. 25 (1978), 161-167. MR 0499081 (58:17043)
  • [19] M. Lipschutz, On strong laws for certain types of events connected with sums of independent random variables, Ann. of Math. 57 (1953), 318-330. MR 0056225 (15:43a)
  • [20] -, On strong bounds for sums of independent random variables which tend to a stable distribution, Trans. Amer. Math. Soc. 81 (1956), 135-154. MR 0077015 (17:979e)
  • [21] -, On the magnitude of the error in the approach of stable distributions. I, II, Indag. Math. 18 (1956), 281-287 and 288-294. MR 0081014 (18:340f)
  • [22] E. Nummelin, A splitting technique for Harris recurrent Markov chains, Z. Wahrsch. Verw. Gebiete 43 (1978), 309-318. MR 0501353 (58:18732)
  • [23] E. Seneta, Regularly varying functions, Lecture Notes Math., vol. 508 Springer, Berlin, Heidelberg and New York, 1976. MR 0453936 (56:12189)
  • [24] M. Thaler, Transformations on $ [0,1]$ with infinite invariant measures, Israel J. Math. 46 (1978), 233-253. MR 727023 (85g:28020)
  • [25] M. J. Wichura, Functional laws of the iterated logarithm for the partial sums of i.i.d. random variables in the domain of attraction of a completely asymmetric stable law, Ann. Probab. 6 (1974), 1108-1138. MR 0358950 (50:11407)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 28D05, 60F15

Retrieve articles in all journals with MSC: 28D05, 60F15


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-1024766-3
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society